A239119 Number of ballot sequences of length n with exactly 8 fixed points.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 9, 29, 99, 357, 1351, 5343, 21992, 93912, 414848, 1891264, 8878972, 42849860, 212214460, 1077052284, 5594301872, 29704267536, 161055535088, 890880956848, 5022885935600, 28843306388880, 168562494708400, 1001888980299056

Offset

0,11

Comments

The fixed points are in the first 8 positions.

Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 8, but not 9. An alternate definition uses the first row.

Conjecture: Generally, for fixed k is column k of A238802 asymptotic to sqrt(2)/(2*(k+1)*(k-1)!) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))), (holds for all k<=10). - Vaclav Kotesovec, Mar 11 2014

Links

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..800

Wikipedia, Young tableau

Formula

See Maple program.

Recurrence (for n>=10): (n-9)*(n^8 - 45*n^7 + 1302*n^6 - 34146*n^5 + 562989*n^4 - 4387005*n^3 + 7242668*n^2 + 80535276*n + 148594320)*a(n) = (n^9 - 54*n^8 + 1392*n^7 - 33705*n^6 + 734286*n^5 - 9696141*n^4 + 60317333*n^3 - 48716460*n^2 - 234532332*n - 4007057040)*a(n-1) + (n-10)*(n-8)*(n^8 - 37*n^7 + 1015*n^6 - 27223*n^5 + 410284*n^4 - 2451988*n^3 - 2863260*n^2 + 83948328*n + 232515360)*a(n-2). - Vaclav Kotesovec, Mar 11 2014

a(n) ~ sqrt(2)/90720 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1+7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 11 2014

Example

a(8) = 1: [1,2,3,4,5,6,7,8].
a(9) = 1: [1,2,3,4,5,6,7,8,1].
a(10) = 3: [1,2,3,4,5,6,7,8,1,1], [1,2,3,4,5,6,7,8,1,2], [1,2,3,4,5,6,7,8,1,9].
a(11) = 9: [1,2,3,4,5,6,7,8,1,1,1], [1,2,3,4,5,6,7,8,1,1,2], [1,2,3,4,5,6,7,8,1,1,9], [1,2,3,4,5,6,7,8,1,2,1], [1,2,3,4,5,6,7,8,1,2,3], [1,2,3,4,5,6,7,8,1,2,9], [1,2,3,4,5,6,7,8,1,9,1], [1,2,3,4,5,6,7,8,1,9,2], [1,2,3,4,5,6,7,8,1,9,10].

Maple

b:= proc(n) option remember; `if`(n<3, [1, 1, 3][n+1],
      ((78*n^4 -18395*n^3 -71700*n^2 +536111*n -6824556)*b(n-1)
       +(203*n^5 +3335*n^4 +113400*n^3 +811949*n^2 -2733405*n
       +5461380)*b(n-2) +(n-3)*(125*n^4 +21309*n^3 +273479*n^2
       +556667*n +1829700)*b(n-3)) /
       (203*n^4+1789*n^3+80693*n^2+377071*n-3156156))
    end:
a:=n-> `if`(n<8, 0, b(n-8)):
seq(a(n), n=0..40);

Mathematica

b[n_, l_List] := b[n, l] = If[n <= 0, 1, b[n - 1, Append[l, 1]] + Sum[If[i == 1 || l[[i - 1]] > l[[i]], b[n - 1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; a[n_] := If[n == 8, 1, b[n - 9, {2, 1, 1, 1, 1, 1, 1, 1}]]; a[n_ /; n < 8] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Crossrefs

Column k=8 of A238802.

Sequence in context: A239116 A239117 A239118 * A238803 A148940 A169781

Adjacent sequences: A239116 A239117 A239118 * A239120 A239121 A239122

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Mar 10 2014

Status

approved